We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. Further, we show that if the ...

In this thesis, we introduce a notion of asymptotic stability for a holomorphic vector bundle with a global holomorphic section on a projective manifold. We prove that the special metric on the bundle studied by Bradlow ...

This thesis presents analytic samplers for the [beta]-Wishart and [beta]-MANOVA ensembles with diagonal covariance. These generalize the [beta]-ensembles of Dumitriu-Edelman, Lippert, Killip-Nenciu, Forrester-Rains, and ...

In this thesis, we investigate the birational geometry of the space of rational curves in various homogeneous spaces, with a focus on the quasi-map compactification induced by the Quot and Hyperquot functors. We first study ...

In this thesis, I study the relationship between the higher rank Donaldson invariants of a smooth 4-manifold X and the invariants of its blowup X#CP2 . This relationship can be expressed in terms of a formal power series ...

In this thesis we apply techniques from the bordered and sutured variants of Floer homology to study Legendrian knots. First, given a front diagram for a Legendrian knot K in S₃ which has been split into several pieces, ...

In the framework of fibred cusp operators on a manifold X associated to a boundary fibration ... , the homotopy groups of the space ... of invertible smoothing perturbations of the identity are computed in terms of the ...

Coalescence of a liquid drop with a liquid bath can be prevented by vibration of the bath. In a certain parameter regime, a purely vertical bouncing motion may ensue. In another, this bouncing state is destabilized by the ...

Abstract: A fundamental problem about irreducible representations of a reductive Lie group G is understanding their restriction to a maximal compact subgroup K. In certain important cases, known as the discrete series, we ...

We analyze the evolution of Laplace eigenvalues on a domain induced by the motion of the boundary. We apply our analysis to two problems: 1. We study the equilibrium and stability of electron bubbles. Electron bubbles are ...

In this thesis, we study the growth of Sobolev norms of global solutions of solutions to nonlinear Schrödinger type equations which we can't bound from above by energy conservation. The growth of such norms gives a ...

The classical Brill-Noether theorems count the dimension of the family of maps from a general curve of genus g to non-degenerate curves of degree d in projective space Pr. These theorems can be extended to include ramification ...

Broken Lefschetz fibrations are a new way to depict smooth 4-manifolds and to investigate their topology; for instance, Perutz defines invariants of 4-manifolds by counting J-holomorphic sections of these fibrations. The ...

This thesis will be concerned with the geometry of minimal submanifolds in certain Riemannian manifolds which possess some special geometric structure. Those Riemannian manifolds will fall into one of the following categories: ...